This direct sum technique has further applications. One application
is to proving a lower bound for approximating the Hamming norm
in a strict turnstile stream.
Using a result of [1]}, to show lower bounds for streaming algorithms in the
strict turnstile model, it suffices to show lower bounds in the simultaneous communication model, where each player simultaneously sends a message to a referee who outputs the answer.
While our direct sum theorem holds in this more restrictive model, we also need to consider a composition of the gap-Hamming problem on top of the Sum-Equal instances as well as an augmented index version of the composed problem.
In the augmented problem
we additionally give a referee an index \(i\)  and the answers to all copies \(j\) , with \(j > i\) .
Similar augmentation has been studied for \(L_p\) -norms [2]}.
This allows us to reduce our communication problem to Hamming norm approximation, and ultimately prove our data stream lower bound.
